(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the set of all n x n matrices A such that AB = BA for a fixed n x n matrix B, is a subspace of M_{nn}.

2. Relevant equations

u+vis in the same vector space asuandv.

kuis in the same vector space asu, wherekis any real number.

3. The attempt at a solution

I am drawn to think of a diagonal matrix when I think of this question. And if I multiply a diagonal matrix by a scalar, it can only be a diagonal matrix or the zero matrix, either way, it AB still equals BA. Similarly, adding two diagonal matrices obtains either another diagonal matrix or the zero matrix...so, in this way, A is a subspace of M_{nn}.

Am I correct?

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# Proving a subspace

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